Adaptive finite element methods based on flux and stress equilibration using FEniCSx

Abstract

This contribution shows how a-posteriori error estimators based on equilibrated fluxes - H(div) functions fulfilling the underlying conservation law - can be implemented in FEniCSx. Therefore, dolfinx_eqlb is introduced, its algorithmic structure is described and classical benchmarks for adaptive solution procedures for the Poisson problem and linear elasticity are presented.

Figures (4)

• Figure 1: Cooks membrane.
• Figure 2: Results of adaptive FEM calculations with different orders $k$ and $m$. E.o.c and $\mathrm{i_{eff}}$ after the final refinement step are reported in (a). The two final meshes for $\kappa_1=100$ are depicted in (b) for $k=m-1=1$ and (c) for $k=m-1=2$.
• Figure 3: Effectivity of the different adaptive solution procedures for the Cooks membrane: (a) summaries the results for the first mesh with $\mathrm{err} = \vert\vert\vert \bm{\mathrm{u}} - \bm{\mathrm{u}}_h \vert\vert\vert \leq 10^{-3}$, (b) details the convergence history (black: $k=2$, blue: $k=3$). Orders $m^*$ indicate the usage of \ref{['eq:elasticity_heuristic-ei']}.
• Figure 4: Performance measurements based on the Cooks membrane. (a) $\mathrm{ratio} = t_\mathrm{eqlb} / t_\mathrm{tot}$ for different primal problems. (b) Accumulated timings using an adaptive algorithm until $\vert\vert\vert \bm{\mathrm{u}} - \bm{\mathrm{u}}_h \vert\vert\vert \leq 10^{-3}$. Orders $m^*$ indicate the use of \ref{['eq:elasticity_heuristic-ei']}.

Theorems & Definitions (6)

• definition 1
• theorem 1
• remark 1
• definition 2
• theorem 2
• proof